A new approach for Solving a nonlinear system of second-order BVPs
Abstract
In this paper, we introduce a new approach based on the Reproducing Kernel Method (RKM) for solving a nonlinear system of second-order Boundary Value Problems (BVPs) without the Gram-Schmidt orthogonalization process. What motivates us to use the RKM without the Gram-Schmidt orthogonalization process is its easy implementation, elimination of the Gram-Schmidt process, fewer calculations, and high accuracy. Finally, the compatibility of numerical results and theorems demonstrates that the Present method is effective.
Keyword : reproducing kernel method, system of second-order boundary value problem, convergence analysis, error analysis
How to Cite
Amoozad, T., Abbasbandy, S., Allahviranloo, T., & Rostamy Malkhalifeh, M. (2024). A new approach for Solving a nonlinear system of second-order BVPs. Mathematical Modelling and Analysis, 29(4), 669–683. https://doi.org/10.3846/mma.2024.19217
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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M. Cakmak and S. Alkan. A numerical method for solving a class of systems of nonlinear Pantograph differential equations. Alexandria Engineering, 61:2651– 2661, 2022. https://doi.org/10.1016/j.aej.2021.07.028
M.G. Cui and Y. Lin. Nonlinear Numerical Analysis in the Reproducing Kernel Space, Nova Science, Hauppauge. Inc., Hauppauge, 2009.
M. Dehghan and A. Saadatmandi. The numerical solution of a nonlinear system of second-order boundary value problems using the sinc-collocation method. Math. Comput. Model., 46:1434–1441, 2007. https://doi.org/10.1016/j.mcm.2007.02.002
A. Faghih and P. Mokhtary. A novel Petrov-Galerkin method for a class of linear systems of fractional differential equations. Appl. Num. Math., 169:396– 414, 2021. https://doi.org/10.1016/j.apnum.2021.07.012
M. Faheem, A. Khan and P.J.Y. Wong. A Legendre wavelet collocation method for 1D and 2D coupled time-fractional nonlinear diffusion system. Comput. Math. Appl., 128:214–238, 2022. https://doi.org/10.1016/j.camwa.2022.10.014
F. Geng and M. Cui. Homotopy perturbation–reproducing kernel method for nonlinear systems of second order boundary value problems. Comput. Appl. Math., 235:2405–2411, 2011. https://doi.org/10.1016/j.cam.2010.10.040
F.Z. Geng and M.G. Cui. Solving a nonlinear system of second order boundary value problems. J. Math. Anal. Appl., 327:1167–1181, 2007. https://doi.org/10.1016/j.jmaa.2006.05.011
F.A. Ghassabzadeh, E. Tohidi and H. Singh. RBF collocation approach to calculate numerically the solution of the nonlinear system of qFDEs. King Saud University-Science, 33:101288, 2021. https://doi.org/10.1016/j.jksus.2020.101288
V.L. Hansen. Functional Analysis, Entering Hilbert Space. World Scientific Publishing Co. Pte. Ltd., 2006. https://doi.org/10.1142/5976
W. Jiang and Z. Chen. Solving a system of linear Volterra integral equations using the new reproducing kernel method. Applied Mathematics and Computation, 219:10225–10230, 2013. https://doi.org/10.1016/j.amc.2013.03.123
B. Liu, Y. Wu, J. Guo, H. Zhang, J. Niu and F. Li. Finite difference Jacobian based Newton-Krylov coupling method for solving multi-physics nonlinear system of nuclear reactor. Annals of Nuclear Energy, 148:107670, 2020. https://doi.org/10.1016/j.anucene.2020.107670
J.F. Lu. Variational iteration method for solving a nonlinear system of secondorder boundary value problems. Comput. Math. Appl., 54:1133–1138, 2007. https://doi.org/10.1016/j.camwa.2006.12.060
J. Niu, M.Q. Xu, Y.Z. Lin and Q. Xue. Numerical solution of nonlinear singular boundary value problems. Comput. Appl. Math., 333:42–51, 2018. https://doi.org/10.1016/j.cam.2017.09.040
J. Niu and J. Zhang. Lobatto-reproducing kernel method for solving a linear system of second order boundary value problems. Appl. Math. Comput, 68:3631– 3653, 2022. https://doi.org/10.1007/s12190-021-01685-9
R. Saadeh. Numerical algorithm to solve a coupled system of fractional order using a novel reproducing kernel method. Alexandria Engineering, 60:4583–4591, 2021. https://doi.org/10.1016/j.aej.2021.03.033
H. Sahihi, T. Allahviranloo and S. Abbasbandy. Solving system of second-order bvps using a new algorithm based on reproducing kernel Hilbert space. Appl. Num. Math., 151:27–39, 2020. https://doi.org/10.1016/j.apnum.2019.12.008
Y. Wang, T. Chaolu and Z. Chen. Using reproducing kernel for solving a class of singular weakly nonlinear boundary value problems. Comput. Appl. Math., 87:367–380, 2010. https://doi.org/10.1080/00207160802047640
Y. Yu, J. Niu, J. Zhang and S.Y. Ning. A reproducing kernel method for nonlinear C-q-fractional IVPs. Appl. Math. Lett, 125:107751J, 2022. https://doi.org/10.1016/j.aml.2021.107751